The Janata Dal-United-Rashtriya Janata Dal and Congress Party combine is reportedly leading in seven of the ten seats that went to polls last week.
The tri-party combine has won four seats and is ahead in three others in the state.
If the leads translate into results, these three parties are likely to ramp up political sentiment against the BJP and its allies in the state, claiming that the latter has limited appeal in the state, and may not be in a position to form the next state government in 2015.
In the May general elections, the BJP won 31 of Bihar's 40 parliamentary seats.
It is expected that Monday's bye-election results will be used by all parties to gauge voter mood ahead of the 2015 state assembly election.